This cheat sheet works for vectors with any number of dimensions. It's sufficient to treat each dimension individually. Each dimension can have different boundaries. A boundary is a periodic interval $[a, b)$ with an inclusive lower bound and an exclusive upper bound.
Wrap Positions
Ensures that positions stay inside the periodic boundaries.
On Generic Boundaries $[a, b)$:
$$x \mathrel{{-}{=}} \text{floor}\left(\frac{x - a}{b - a}\right) \cdot (b - a)$$Special Cases
On Boundaries $[0, 1)$:
$$x \mathrel{{-}{=}} \text{floor}(x)$$On Boundaries $[0, w)$:
$$x \mathrel{{-}{=}} \text{floor}(x/w) \cdot w$$On Boundaries $[-v, v)$:
$$x \mathrel{{-}{=}} \text{round}\left(\frac{x}{2v}\right) \cdot 2v$$Shortest Connection
Determine the shortest connection of two positions in your periodic space. The connection can wrap around boundaries. First, compute the connection in the usual way:
$$dx = x_2 - x_1$$Then, wrap the connection:
On Generic Boundaries $[a, b)$:
$$dx \mathrel{{-}{=}} \text{round}\left(\frac{dx}{b - a}\right) \cdot (b - a)$$Special Cases:
On Boundaries $[0, 1)$:
$$dx \mathrel{{-}{=}} \text{round}(dx)$$